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What do you understand by intramodal dispersion? Derive the expression for material dispersion.
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• Pulse broadening within a single mode is called as intramodal dispersion or chromatic dispersion.

• The two main causes of intramodal dispersion are as follows:

a. Waveguide dispersion:

• It occurs because a single mode fiber confines only about 80% of the optical power to the core.

• Dispersion thus arises since the 20% light propagating in the cladding travels faster than light confined to the core.

b. Material dispersion:

• It is the pulse spreading due to the dispersive properties of material.

• It arises from variation of refractive index of the core material as a function of wavelength.

• Material dispersion is a property of glass as a material and will always exist irrespective of the structure of the fiber.

• It occurs when the phase velocity of the plane wave propagation in the dielectric medium varies non-linearly with wavelength and a material is said to exhibit a material dispersion, when the second differential of the Refractive index w.r.t wavelength is not zero.

i.e. $\frac{d^2 n}{dλ^2} ≠ 0$

• The pulse spread due to material dispersion may be obtained by considering the group delay $τ_g$ in the optical fiber which is the reciprocal of group velocity $v_g$. The group delay is given by

$$τ_g= \frac{d\beta}{d\omega} = 1/c (n_1-\frac{λdn_1}{dλ})-------(1)$$

where $n_1$ is the refractive index of the core material

$ω$ is the angular frequency $\beta$ is the propagation constant\lt/constant\gt The pulse delay $τ_m$ due to material dispersion in a fiber of length L is $$τ_m = \frac{L}C (n_1-λ \frac{dn_1}{dλ}) ------------- (2)$$

For a source with rms spectral width $σ_λ$ & mean wavelength λ, the rms pulse broadening due to material dispersion $σ_m$ may be obtained from the expansion of equation (2) in a Taylor series about λ.

$$σ_m = σ_λ \frac{dτ_m}{dλ} + σ_λ \frac{d^2 τ_m}{dλ^2} + ------------- (3)$$

As the 1st term in eq.(3) usually dominate for the source operating over 0.8-0.9 μm wavelength range.

$$σ_m = σ_λ \frac{dτ_m}{d_λ} --------------- (4)$$

Hence the pulse Spread may be evaluated by considering the dependence of $τ_m$ on λ. From eq.(2)

$\frac{dτ_m}{dλ} = L \fracλ{C} \bigg[\frac{dn_1}{dλ} - \frac{d^2 n_1}{dλ^2} - \frac{dn_1}{dλ}\bigg]\\ = -\frac{Lλ}{C} \frac{d^2 n_1}{dλ^2}--------(5)$

Substitute eqn(5) in eqn(4)

The rms pulse broadening due to material dispersion is given by

$$σ_m = \frac{σ_λ L}C│λ \frac{d^2 n_1}{dλ^2}│ ---------- (6)$$

The material dispersion for optical fiber is sometimes quoted as the $│λ^2 \frac{d^2 n_1}{dλ^2}^)│$

or $│d^2 n_1/dλ^2│$

However it may be given in terms of material dispersion parameter M given as:

$$M = \frac1{L} \frac{dτ_m}{dλ} = \frac{λ}C│\frac{d^2 n_1}{dλ^2} │$$

Total pulse spreading caused by material dispersion is given by $∆_{mat}$ (P.S)

where $\Deltaλ$ is the spectral width of light source

L is the fiber length

$$∆t_{mat} (P.S) = M.L (∆λ)$$